Calculations of single particle spectra in density functional theory

In the context of the density functional theory we consider the single particle excitation spectra of electron systems. As a result, we have related the single particle excitations with the eigenvalues of the corresponding Kohn-Sham equations. In a case when the exchange correlation functional is approximated by the exchange functional the coupling equations are very simple, while the single particle spectra do not coincide neither with the eigenvalues of the Kohn-Sham equations nor with the ones of the Hartree-Fock equations.